# Standard normal covariance

I am given that X is a standard normal distribution.

Why is $$Cov(X, -X) = -1$$?

I know that $$Cov(X,X) = Var(X)$$ and that the $$Var(X) = 1$$.

Is the $$Var(-X) = -1$$?

This follows from $$Cov(X,-X)=-Cov(X,X)$$. In fact, more generally, $$Cov(aX,bY)=abCov(X,Y)$$.
Also, the above property can be used to show $$Var(-X)=(-1)^2Var(X)=Var(X)$$ so $$Var(-X)\neq -1$$ (in fact, the variance is always nonnegative).